Non-equilibrium geosystems: analyzing Lyapunov concepts on pattern formation

In geomorphology literature, equilibrium refers to researches of Davis and Gilbert who presented steady-state and dynamics equilibrium. To analyze chaotic geosystems and pattern formation in non-equlibrium conditions, the nonlinear thermodynamics concepts is needed. In this article, have been analyzed geosystems responses in non-equilibrium states by Lyapunov theory. Self-organization of dissipative systems leads to form regular pattern in non-equilibrium which can be a sign to forecast geosystem evolutions. Non-Equilibrium Geosystems: The second law of thermodynamics asserts that if a spontaneous reaction occurs, the reaction moves towards an irreversible state of equilibrium and in the process, becomes increasingly random or disordered. It is this increasing disorder or entropy of a system that forces a spontaneous reaction to persist; but, once a system attains maximum entropy or equilibrium, the spontaneous reaction ceases to continue.Non-equilibrium systems are maintained in a state away from thermodynamic equilibrium by the steady injection and transport of energy. Most interesting to us is systems displaying regular or nearly regular spatial structures same ripple marks and river pattern in geosystems.A particular nonequilibrium system can be thought of as occupying a point in a three-dimensional parameter space with axes labeled by three dimensionless parameters R, Γ, and N.The parameter R is some dimensionless parameter like the Rayleigh number that measures the strength of driving compared to dissipation. For many systems, driving a system further from equilibrium by increasing R to larger values leads to chaos and then to ever-more complicated spatiotemporal states for which there is ever finer spatial structure and ever faster temporal dynamics. Lyapunov theory and regular pattern: An n-dimensional (where the number of dimensions equals the number of components) system has n Lyapunov exponents, which determine the rate of convergence or divergence of initially similar system states in the system phase space, and thus the sensitivity to perturbations or to variations in initial conditions.The system is not, and cannot be, chaotic unless there is at least one positive Lyapunov exponent. Because an unstable system has at least one λ > 0, dynamic instability is tantamount to a chaotic system. Deterministic chaos is a property of some nonlinear systems whereby even simple deterministic systems can produce complex, pseudorandom patterns, independently of stochastic forcing or environmental heterogeneity. In chaotic systems complexity and unpredictability are inherent in system dynamics. Such systems are strongly sensitive to initial conditions, in that initially similar states diverge exponentially, on average, and become increasingly different over time. Chaotic systems are also sensitive to perturbations of all magnitudes.The Kolmogorov (K-) entropy of a nonlinear system measures its 'chaoticity', because K-entropy is equal to the sum of the positive Lyapunov exponents. In real landscapes, measured entropy can be due to deterministic complexity, or to 'colored noise', the combination of randomness and deterministic order. Culling (1988b) was apparently the first to suggest exploiting the relationship between K-entropy (estimated using standard statistical or information theoretic entropy measures) and chaos in geomorphic systems. There are three forms of entropy referred to in geomorphology. Thermodynamic entropy is a measure of the amount of thermal energy unavailable to do work, or the disorder in a closed system. Statistical (information theoretic) entropy measures the loss of information in a transmission, or the degree of disorder in a statistical distribution.Kolmogorov (K-) entropy measures the expansion of a system's phase space (the n-dimensional space defining all possible system states or combinations of values of the n components). Many geomorphic systems show clear evidence of chaotic dynamics and deterministic complexity. These phenomena cause many nonlinear dynamical systems to behave unpredictably (at certain scales), to exhibit extraordinary sensitivity to initial conditions, and to show complicated, pseudorandom patterns even in the absence of environmental heterogeneity and stochastic forcing. The positive λ reflect the K-entropy or 'chaoticity', and the rate of disorganization; the negative λ give the rate of organization. If an open, dissipative geomorphic system is to organize itself, there must be at least one positive Lyapunov exponent, but the sum of λ must be negative. The sum of the diagonal elements of the system interaction matrix is equal to the sum of real parts of the complex eigenvalues, and to the Lyapunov exponents.